Base field 6.6.1241125.1
Generator \(w\), with minimal polynomial \(x^{6} - 7x^{4} - 2x^{3} + 11x^{2} + 7x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[45, 15, 2w^{5} - 14w^{3} - 3w^{2} + 23w + 9]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $27$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} - 74x^{5} - 40x^{4} + 1744x^{3} + 2112x^{2} - 13248x - 24192\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -2w^{5} + w^{4} + 13w^{3} - 2w^{2} - 19w - 5]$ | $-1$ |
| 9 | $[9, 3, 2w^{5} - w^{4} - 14w^{3} + 2w^{2} + 23w + 6]$ | $-1$ |
| 11 | $[11, 11, w - 1]$ | $\phantom{-}e$ |
| 25 | $[25, 5, w^{3} + w^{2} - 4w - 3]$ | $-\frac{1}{96}e^{6} + \frac{1}{16}e^{5} + \frac{37}{48}e^{4} - \frac{77}{24}e^{3} - \frac{109}{6}e^{2} + 37e + 142$ |
| 29 | $[29, 29, w^{5} - w^{4} - 7w^{3} + 4w^{2} + 11w]$ | $\phantom{-}\frac{1}{2}e^{2} - 12$ |
| 41 | $[41, 41, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $\phantom{-}\frac{1}{120}e^{6} + \frac{1}{40}e^{5} - \frac{5}{12}e^{4} - \frac{13}{12}e^{3} + \frac{151}{30}e^{2} + \frac{51}{5}e - \frac{54}{5}$ |
| 49 | $[49, 7, w^{5} - w^{4} - 7w^{3} + 4w^{2} + 11w + 1]$ | $\phantom{-}\frac{1}{480}e^{6} + \frac{3}{80}e^{5} - \frac{5}{48}e^{4} - \frac{47}{24}e^{3} + \frac{19}{30}e^{2} + \frac{114}{5}e + \frac{104}{5}$ |
| 59 | $[59, 59, 2w^{5} - w^{4} - 14w^{3} + 2w^{2} + 24w + 7]$ | $-\frac{1}{40}e^{6} + \frac{1}{20}e^{5} + \frac{3}{2}e^{4} - \frac{5}{2}e^{3} - \frac{261}{10}e^{2} + \frac{127}{5}e + \frac{732}{5}$ |
| 59 | $[59, 59, -w^{5} + 8w^{3} + 2w^{2} - 15w - 8]$ | $-\frac{3}{160}e^{6} + \frac{3}{80}e^{5} + \frac{19}{16}e^{4} - \frac{17}{8}e^{3} - \frac{227}{10}e^{2} + \frac{129}{5}e + \frac{714}{5}$ |
| 61 | $[61, 61, w^{5} - 7w^{3} - 2w^{2} + 12w + 4]$ | $-\frac{1}{48}e^{6} + \frac{1}{8}e^{5} + \frac{37}{24}e^{4} - \frac{77}{12}e^{3} - \frac{215}{6}e^{2} + 73e + 280$ |
| 61 | $[61, 61, -w^{5} + 7w^{3} - 11w - 1]$ | $\phantom{-}\frac{1}{80}e^{6} - \frac{1}{40}e^{5} - \frac{7}{8}e^{4} + \frac{5}{4}e^{3} + \frac{183}{10}e^{2} - \frac{66}{5}e - \frac{556}{5}$ |
| 64 | $[64, 2, 2]$ | $-\frac{1}{96}e^{6} + \frac{1}{8}e^{5} + \frac{49}{48}e^{4} - \frac{19}{3}e^{3} - \frac{89}{3}e^{2} + 72e + 257$ |
| 71 | $[71, 71, w^{3} + w^{2} - 5w - 3]$ | $\phantom{-}\frac{1}{8}e^{5} + \frac{1}{4}e^{4} - \frac{25}{4}e^{3} - \frac{25}{2}e^{2} + 69e + 156$ |
| 71 | $[71, 71, -3w^{5} + w^{4} + 21w^{3} - w^{2} - 33w - 9]$ | $\phantom{-}\frac{1}{60}e^{6} - \frac{3}{40}e^{5} - \frac{13}{12}e^{4} + \frac{43}{12}e^{3} + \frac{316}{15}e^{2} - \frac{178}{5}e - \frac{648}{5}$ |
| 79 | $[79, 79, -2w^{5} + w^{4} + 13w^{3} - 3w^{2} - 19w - 5]$ | $\phantom{-}\frac{1}{30}e^{6} - \frac{1}{40}e^{5} - \frac{23}{12}e^{4} + \frac{17}{12}e^{3} + \frac{919}{30}e^{2} - \frac{71}{5}e - \frac{706}{5}$ |
| 81 | $[81, 3, 2w^{5} - w^{4} - 13w^{3} + w^{2} + 19w + 8]$ | $\phantom{-}\frac{17}{480}e^{6} - \frac{9}{80}e^{5} - \frac{109}{48}e^{4} + \frac{137}{24}e^{3} + \frac{664}{15}e^{2} - \frac{302}{5}e - \frac{1412}{5}$ |
| 89 | $[89, 89, 2w^{5} - w^{4} - 13w^{3} + 2w^{2} + 20w + 7]$ | $-\frac{7}{240}e^{6} - \frac{1}{40}e^{5} + \frac{41}{24}e^{4} + \frac{11}{12}e^{3} - \frac{418}{15}e^{2} - \frac{36}{5}e + \frac{594}{5}$ |
| 89 | $[89, 89, -w^{5} + 8w^{3} + w^{2} - 16w - 5]$ | $-\frac{1}{120}e^{6} - \frac{1}{40}e^{5} + \frac{5}{12}e^{4} + \frac{13}{12}e^{3} - \frac{83}{15}e^{2} - \frac{46}{5}e + \frac{114}{5}$ |
| 89 | $[89, 89, -3w^{5} + w^{4} + 20w^{3} - 30w - 11]$ | $-\frac{1}{240}e^{6} + \frac{1}{20}e^{5} + \frac{11}{24}e^{4} - \frac{7}{3}e^{3} - \frac{413}{30}e^{2} + \frac{117}{5}e + \frac{582}{5}$ |
| 89 | $[89, 89, -w^{5} + 7w^{3} + 2w^{2} - 11w - 4]$ | $-\frac{3}{160}e^{6} + \frac{3}{80}e^{5} + \frac{19}{16}e^{4} - \frac{17}{8}e^{3} - \frac{116}{5}e^{2} + \frac{139}{5}e + \frac{744}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $5$ | $[5, 5, -2w^{5} + w^{4} + 13w^{3} - 2w^{2} - 19w - 5]$ | $1$ |
| $9$ | $[9, 3, 2w^{5} - w^{4} - 14w^{3} + 2w^{2} + 23w + 6]$ | $1$ |